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URL: https://oeis.org/A290173

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A290173
Bases b for which there exists an integer y such that y^2 in base b looks like [c,d,c,d,c,d] for some base-b digits c, d.
4
68, 313, 699, 4366, 51567, 234924, 686287, 3526450, 3652434, 301121223, 1250094151, 3072665429, 4507947478
OFFSET
1,1
COMMENTS
Equivalently, numbers k such that A007913(1 + k^2 + k^4) < k^2, where A007913(n) is the squarefree part of n. - Giovanni Resta, Jul 27 2017
REFERENCES
Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, Experimental Math, 28 (2019), 428-439.
LINKS
Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, preprint, arXiv:1707.03894 [math.NT], July 14 2017.
EXAMPLE
For example, for b = 68, we have y = 160797, and the base-b representation of y^2 is (17,53,17,53,17,53).
MATHEMATICA
core[n_] := Block[{f = Transpose@ FactorInteger@ n}, Times @@ (f[[1]]^ Mod[f[[2]], 2])]; Select[Range[3*10^5], core[1 + # + #^2] core[1 - # + #^2] < #^2 &] (* Giovanni Resta, Jul 27 2017 *)
PROG
(PARI) isok(k) = core(1+k^2+k^4) < k^2; \\ Michel Marcus, Jul 28 2017
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
Jeffrey Shallit, Jul 23 2017
EXTENSIONS
a(10)-a(13) from Giovanni Resta, Jul 27 2017
STATUS
approved